Serway & Jewett — Circular Motion and Other Applications of Newton's Laws: Problem 39

Galileo thought about whether acceleration should be defined as the rate of change of velocity over time or as the rate of change in velocity over distance. He chose the former, so let’s use the name “vroomosity” for the rate of change of velocity over distance. For motion of a particle on a straight line with constant acceleration, the equation \(v = v_i + at\) gives its velocity \(v\) as a function of time. Similarly, for a particle’s linear motion with constant vroomosity \(k\), the equation \(v = v_i + kx\) gives the velocity as a function of the position \(x\) if the particle’s speed is \(v_i\) at \(x = 0\). (a) Find the law describing the total force acting on this object of mass \(m\). Describe an example of such a motion or explain why it is unrealistic for (b) the possibility of \(k\) positive and (c) the possibility of \(k\) negative.